Problem: Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$.
Explanation: By Pascal's Identity, we have $\binom{23}{4} + \binom{23}{5} = \binom{24}{5}$.  However, we also have $\binom{24}{5} = \binom{24}{24-5} = \binom{24}{19}$.  There are no other values of $k$ such that $\binom{24}{5} = \binom{24}{k}$, so the sum of all integers that satisfy the problem is $5+19 = \boxed{24}$.

Challenge: Is it a coincidence that the answer is 24?